Lemma 37.61.4. A composition of perfect morphisms of schemes is perfect.
Proof. This translates into the following algebra result: If A \to B \to C are composable perfect ring maps then A \to C is perfect. We have seen this is the case for pseudo-coherent in Lemma 37.60.4 and its proof. By assumption there exist integers n, m such that B has tor dimension \leq n over A and C has tor dimension \leq m over B. Then for any A-module M we have
M \otimes _ A^{\mathbf{L}} C = (M \otimes _ A^{\mathbf{L}} B) \otimes _ B^{\mathbf{L}} C
and the spectral sequence of More on Algebra, Example 15.62.4 shows that \text{Tor}^ A_ p(M, C) = 0 for p > n + m as desired. \square
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